Metamaterial measurement in a cylindrical coaxial fixture with consideration for inter-element coupling

Summary form given only: A great deal of research has been dedicated to the design and synthesis of metamaterials to acheive interesting electromagnetic properties such as very high, sub-unity, and even negative effective relative permeabilities and permittivities. While the design and fabrication of various metamaterial structures can be a challenge, it can be equally demanding in some cases to make accurate measurements that properly characterize the metamaterial. It has been shown that coaxial measurements of metamaterial structures can provide appealing advantages over other approaches although in some cases, it may be necessary to modify the metamaterial depending on its structure or mode of excitation. Fortunately, for planar metamaterials designed to operate under approximately TEM wave excitation, a coaxial cell can work quite nicely. It has, however, been discussed that inter-element coupling along with coupling to the waveguide walls can cause discrepancies between the behavior of the metamaterial in a waveguide and its behavior in an intended application. The Lagrangian formalism for metamaterials has been used extensively in recent years to describe the coupling effects between resonant metamaterial structures and predict the shift of the resonant frequency in the presence of coupling. The Lagrangian formulation is modified here and used to describe the coupling effects of metamaterials near curved conductors such as in the coaxial case. Thus we are able to account for the coupling effects due the curved waveguide walls and predict the shift in the resonant frequency between measurements in a coaxial fixture and other scenarios, such as free space for example. To measure metamaterial parameters, a coaxial fixture along with a complete TRL fabrication kit was fabricated with 3d printing technology. The fixture was made of solid ABS plastic with an inner radius of a = 3.175mm, an outer radius of b = 26.675mm, a dielectric constant of Er ≈ 3 and a loss tangent of tan δ ≈ 0.05 at 1.5 GHz. To transition from the 50Ω SMA measurement cables to the fixture with a characteristic impedance of approximately Z0 ≈ 76Ω, tapered clam-shell transition sections were used. Through measurements of the unloaded fixture, it was apparent that higher order modes began to propagate at approximately 1.5 GHz indicating the limit of the useful bandwidth. To validate the fixture, a metamaterial sample composed of a set of 3 split ring resonators were measured. After calibration, measured data matched very well with simulated data from HFSS with ideal waveports.